No, there isn't a need for a naive controller driver to be part of steam. So much so that in fact it isn't, the driver on Linux is called hid-steam and it's open source and was developed with help from Valve, which is why the controller works even outside of Steam.

What you keep missing is that the default behavior is purposefully that of a mouse+KB because it's what makes sense. Go ahead and plug the controller on a Linux machine and you will be able to use it as a mouse, click on a game to launch it, then it will become a controller when the game tries to grab it (unfortunately some games try to be smart about this and fail miserably and don't detect it), but a naive one, similar to the Xbox controller, you'll miss the back buttons, touchpads and extra sensors because most controllers don't have them, and without SteamInput to remap those on the fly you'll be left with whatever the game decides to handle, which is usually just an Xbox controller. That SDL thing you linked gives a way for games to handle those extra inputs, but it's still relying on the open source driver that's already there, and it depends on the game to use it.

You're not asking for a driver, you have that already, you're asking for SteamInput to be released separately from Steam, which is a weird ask. SteamInput is a product that Valve develops to bring people to Steam and allows to remap any controller to any input, nothing stops people from making an alternative to it, but expecting Valve to release it separately is weird.